Q. Consider 𝑝(𝑠) = 𝑠^{3} + 𝑎_{2}𝑠^{2} + 𝑎_{1}𝑠 + 𝑎_{0} with all real coefficients. It is known that its derivative 𝑝′(𝑠) has no real roots. The number of real roots of 𝑝(𝑠) is

(A) 0 (B) 1 (C) 2 (D) 3

Ans: 1

Sol:

Given p(s) = s3 + a2 s2 + a1s + a0

p'(s) = 3s2 + 2a2s + a1

Also given that p'(s) has no real roots, therefore, we can write:

n – 1 = 0

n = Number of real roots in p(s)

∴ n = 1